N(t)=500e^(kt)

N(3) = 1200

500e^(k*3) = 1200

First we will find k by solving:

500e^(k*3) = 1200 divide both sides by 500

e^(3k) = 2.4

(e^k)^3 = 2.4

Let t represent the time in hours needed for the original population to double in size:

N(t) = 2N(0)

500e^(kt) = 2*500e^(k*0)

500e^(kt) = 1000 divide both sides by 500

(e^k)^t = 2

(e^k)^(3t) = 2^3

((e^k)^3)^t = 8

2.4^t = 8

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t = 2.38 hr

The time required for the original population to double in size is 2.38 hours.